Chapter Outline: Phase Diagrams
Microstructure and Phase Transformations in
Multicomponent Systems
ƒ Definitions and basic concepts
ƒ Phases and microstructure
ƒ Binary isomorphous systems (complete solid solubility)
ƒ Binary eutectic systems (limited solid solubility)
ƒ Binary systems with intermediate phases/compounds
ƒ The iron-carbon system (steel and cast iron)
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Chapter 9, Phase Diagrams
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Definitions: Components and Phases
Component - chemically recognizable species (Fe and C
in carbon steel, H2O and Sucrose in sugar solution in
water). A binary alloy contains two components, a ternary
alloy – three, etc.
Phase – a portion of a system that has uniform physical
and chemical characteristics. Two distinct phases in a
system have distinct physical and/or chemical
characteristics (e.g. water and ice, water and oil) and are
separated from each other by definite phase boundaries. A
phase may contain one or more components.
A single-phase system is called homogeneous,
systems with two or more phases are mixtures or
heterogeneous systems.
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Definitions: Solubility Limit
Solvent - host or major component in solution, solute minor component.
Solubility Limit of a component in a phase is the
maximum amount of the component that can be dissolved
in it (e.g. alcohol has unlimited solubility in water, sugar
has a limited solubility, oil is insoluble). The same
concepts apply to solid phases: Cu and Ni are mutually
soluble in any amount (unlimited solid solubility), while C
has a limited solubility in Fe.
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Microstructure
The properties of an alloy depend not only on proportions
of the phases but also on how they are arranged structurally
at the microscopic level. Thus, the microstructure is
specified by the number of phases, their proportions, and
their arrangement in space.
Microstructure of cast Iron
http://www2.umist.ac.uk/material/research/intmic/ (link is dead)
The long gray regions are flakes of graphite.
The matrix is a fine mixture of BCC Fe and Fe3C compound.
Phase diagrams will help us to understand and predict
microstructures like the one shown in this page
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Equilibrium and Metastable States
A system is at equilibrium if at constant temperature,
pressure and composition the system is stable, not
changing with time.
Equilibrium is the state that is achieved given sufficient
time. But the time to achieve equilibrium may be very long
(the kinetics can be slow) that a state along the path to the
equilibrium may appear to be stable. This is called a
metastable state.
In thermodynamics the equilibrium is described as a state of a
system that corresponds to the minimum of thermodynamic
function called the free energy. Thermodynamics tells us that:
• The state of stable thermodynamic
equilibrium is the one with
minimum free energy.
• A system at a metastable state is
trapped in a local minimum of free
energy that is not the global one.
Free Energy
• Under conditions of a constant temperature and pressure and
composition, the direction of any spontaneous change is
toward a lower free energy.
equilibrium
metastable
Arrangement of atoms
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Phase diagram
Phase diagram is a graphical representation of all the
equilibrium phases as a function of temperature, pressure,
and composition.
For one component systems, the equilibrium state of the
system is defined by two independent parameters (P and
T), (T and V), or (P and V).
Pressure-temperature phase diagram for H2O:
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Chapter 9, Phase Diagrams
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PVT surface of a pure (1-component) substance
http://www.eng.usf.edu/~campbell/ThermoI/ThermoI_mod.html
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A pure substance is heated at constant pressure
T
Tb
V
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Pressure-temperature phase diagram for carbon
We can see graphite, diamond, liquid carbon on the phase
diagram… but where are fullerenes and nanotubes?
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Phase diagrams for binary systems
A phase diagrams show what phases exist at equilibrium
and what phase transformations we can expect when we
change one of the parameters of the system.
Real materials are almost always mixtures of different
elements rather than pure substances: in addition to T and
P, composition is also a variable.
We will limit our discussion of phase diagrams of multicomponent systems to binary alloys and will assume
pressure to be constant at one atmosphere. Phase diagrams
for materials with more than two components are complex
and difficult to represent. An example of a phase diagram
for a ternary alloy is shown for a fixed T and P below.
ternary phase diagram
of Ni-Cr-Fe
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Binary Isomorphous Systems (I)
Isomorphous system - complete solid solubility of the two
components (both in the liquid and solid phases).
L
α+L
α
Three phase region can be identified on the phase diagram:
Liquid (L) , solid + liquid (α +L), solid (α )
Liquidus line separates liquid from liquid + solid
Solidus line separates solid from liquid + solid
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Binary Isomorphous Systems (II)
Example of isomorphous system: Cu-Ni (the complete
solubility occurs because both Cu and Ni have the same
crystal structure, FCC, similar radii and electronegativity).
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Binary Isomorphous Systems (III)
In one-component system melting occurs at a well-defined
melting temperature.
In multi-component systems melting occurs over the range
of temperatures, between the solidus and liquidus lines.
Solid and liquid phases are at equilibrium with each other
in this temperature range.
Liquidus
Temperature
L
liquid solution
α+L
Solidus
liquid solution
+
crystallites of
solid solution
α
polycrystal
solid solution
A
20 40 60 80
Composition, wt %
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B
Chapter 9, Phase Diagrams
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Interpretation of a binary phase diagrams
For a given temperature and composition we can use phase
diagram to determine:
1) The phases that are present
2) Compositions of the phases
3) The relative fractions of the phases
Finding the composition in a two phase region:
1. Locate composition and temperature in diagram
2. In two phase region draw the tie line or isotherm
3. Note intersection with phase boundaries. Read
compositions at the intersections.
The liquid and solid phases have these compositions.
XB
Xliquid
B
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Xsolid
B
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The lever rule
Finding the amounts of phases in a two phase region:
1. Locate composition and temperature in diagram
2. In two phase region draw the tie line or isotherm
3. Fraction of a phase is determined by taking the length
of the tie line to the phase boundary for the other
phase, and dividing by the total length of tie line
α+β
α
β
Wα
Wβ
The lever rule is a mechanical analogy to the mass balance
calculation. The tie line in the two-phase region is
analogous to a lever balanced on a fulcrum.
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Derivation of the lever rule
1) All material must be in one phase or the other:
Wα + WL = 1
2) Mass of a component that is present in both phases
equal to the mass of the component in one phase +
mass of the component in the second phase:
WαCα + WLCL = Co
3) Solution of these equations gives us the lever rule.
WL = (Cα - Co) / (Cα - CL)
Wα = (Co - CL) / (Cα - CL)
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Phase compositions and amounts. An example.
Co = 35 wt. %, CL = 31.5 wt. %, Cα = 42.5 wt. %
Mass fractions:
WL = S / (R+S) = (Cα - Co) / (Cα - CL) = 0.68
Wα = R / (R+S) = (Co - CL) / (Cα - CL) = 0.32
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Development of microstructure in isomorphous alloys
Equilibrium (very slow) cooling
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Development of microstructure in isomorphous alloys
Equilibrium (very slow) cooling
¾ Solidification in the solid + liquid phase occurs
gradually upon cooling from the liquidus line.
¾ The composition of the solid and the liquid change
gradually during cooling (as can be determined by the
tie-line method.)
¾ Nuclei of the solid phase form and they grow to
consume all the liquid at the solidus line.
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Development of microstructure in isomorphous alloys
Fast (non-equilibrium) cooling
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Development of microstructure in isomorphous alloys
Fast (non-equilibrium) cooling
• Compositional changes require diffusion in solid and
liquid phases
• Diffusion in the solid state is very slow. ⇒ The new
layers that solidify on top of the existing grains have the
equilibrium composition at that temperature but once
they are solid their composition does not change. ⇒
Formation of layered (cored) grains and the invalidity of
the tie-line method to determine the composition of the
solid phase.
• The tie-line method still works for the liquid phase,
where diffusion is fast. Average Ni content of solid
grains is higher. ⇒ Application of the lever rule gives
us a greater proportion of liquid phase as compared to
the one for equilibrium cooling at the same T. ⇒
Solidus line is shifted to the right (higher Ni contents),
solidification is complete at lower T, the outer part of
the grains are richer in the low-melting component (Cu).
• Upon heating grain boundaries will melt first. This can
lead to premature mechanical failure.
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Binary Eutectic Systems (I)
systems (alloys) with limited solubility
Credit: Kenneth A. Jackson, University of Arizona.
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Binary Eutectic Systems (II)
Liquidus
liquid
Temperature, °C
Solidus
Solvus
α+β
Copper – Silver phase diagram
Composition, wt% Ag
Three single phase regions (α - solid solution of Ag in Cu
matrix, β = solid solution of Cu in Ag matrix, L - liquid)
Three two-phase regions (α + L, β +L, α +β)
Solvus line separates one solid solution from a mixture of
solid solutions. Solvus line shows limit of solubility
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Binary Eutectic Systems (III)
Temperature, °C
Lead – Tin phase diagram
Invariant or eutectic point
Eutectic isotherm
Composition, wt% Sn
Eutectic or invariant point - Liquid and two solid phases
co-exist in equilibrium at the eutectic composition CE and
the eutectic temperature TE.
Eutectic isotherm - the horizontal solidus line at TE.
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Binary Eutectic Systems (IV)
Eutectic reaction – transition between liquid and mixture
of two solid phases, α + β at eutectic concentration CE.
The melting point of the eutectic alloy is lower than that of
the components (eutectic = easy to melt in Greek).
At most two phases can be in equilibrium within a phase
field. Three phases (L, α, β) may be in equilibrium only
at a few points along the eutectic isotherm. Single-phase
regions are separated by 2-phase regions.
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Binary Eutectic Systems (V)
Compositions and relative amounts of phases are
determined from the same tie lines and lever rule, as for
isomorphous alloys
•A
•B
•C
For points A, B, and C calculate the compositions (wt. %)
and relative amounts (mass fractions) of phases present.
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Development of microstructure in eutectic alloys (I)
Several different types of microstructure can be formed in
slow cooling an different compositions.
Let’s consider cooling of liquid lead – tin system at
different compositions.
In this case of lead-rich
alloy (0-2 wt. % of tin)
solidification proceeds in
the same manner as for
isomorphous alloys (e.g.
Cu-Ni) that we discussed
earlier.
L → α +L→ α
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Development of microstructure in eutectic alloys (II)
At compositions between the room temperature solubility
limit and the maximum solid solubility at the eutectic
temperature, β phase nucleates as the α solid solubility is
exceeded upon crossing the solvus line.
L
α +L
α
α +β
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Development of microstructure in eutectic alloys (III)
Solidification at the eutectic composition
No changes above the eutectic temperature TE. At TE the
liquid transforms to α and β phases (eutectic reaction).
L → α +β
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Development of microstructure in eutectic alloys (IV)
Solidification at the eutectic composition
Compositions of α and β phases are very different →
eutectic reaction involves redistribution of Pb and Sn atoms
by atomic diffusion. This simultaneous formation of α and
β phases result in a layered (lamellar) microstructure that is
called eutectic structure.
Formation of the eutectic structure in the lead-tin system.
In the micrograph, the dark layers are lead-reach α phase, the
light layers are the tin-reach β phase.
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Development of microstructure in eutectic alloys (V)
Compositions other than eutectic but within the range of
the eutectic isotherm
Primary α phase is formed in the α + L region, and the
eutectic structure that includes layers of α and β phases
(called eutectic α and eutectic β phases) is formed upon
crossing the eutectic isotherm.
L → α + L → α +β
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Development of microstructure in eutectic alloys (VI)
Microconstituent – element of the microstructure having a
distinctive structure. In the case described in the previous
page, microstructure consists of two microconstituents,
primary α phase and the eutectic structure.
Although the eutectic structure consists of two phases, it is
a microconstituent with distinct lamellar structure and
fixed ratio of the two phases.
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How to calculate relative amounts of microconstituents?
Eutectic microconstituent forms from liquid having eutectic
composition (61.9 wt% Sn)
We can treat the eutectic as a separate phase and apply the
lever rule to find the relative fractions of primary α phase
(18.3 wt% Sn) and the eutectic structure (61.9 wt% Sn):
We = P / (P+Q) (eutectic)
MSE 2090: Introduction to Materials Science
Wα’ = Q / (P+Q) (primary)
Chapter 9, Phase Diagrams
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How to calculate the total amounts of phases?
Fraction of α phase determined by application of the lever
rule across the entire α + β phase field:
Wα = (Q+R) / (P+Q+R) (α phase)
Wβ = P / (P+Q+R) (β phase)
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Phase diagrams with intermediate phases
Eutectic systems that we have studied so far have only two
solid phases (α and β) that exist near the ends of phase
diagrams.
These phases are called terminal solid
solutions.
Some binary alloy systems have intermediate solid
solution phases. In phase diagrams, these phases are
separated from the composition extremes (0% and 100%).
Example: in Cu-Zn, α and η are terminal solid solutions,
β, β’, γ, δ, ε are intermediate solid solutions.
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Phase diagrams with intermetallic compounds
Besides solid solutions, intermetallic compounds, that
have precise chemical compositions can exist in some
systems.
When using the lever rules, intermetallic compounds are
treated like any other phase, except they appear not as a
wide region but as a vertical line.
intermetallic
compound
This diagram can be thought of as two joined eutectic
diagrams, for Mg-Mg2Pb and Mg2Pb-Pb. In this case
compound Mg2Pb can be considered as a component.
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Eutectoid Reactions (I)
The eutectoid (eutectic-like in Greek) reaction is similar to
the eutectic reaction but occurs from one solid phase to two
new solid phases.
Invariant point (the eutectoid) – three solid phases are in
equilibrium.
Upon cooling, a solid phase transforms into two other solid
phases (δ ↔ γ + ε in the example below)
Looks as V on top of a horizontal tie line (eutectoid
isotherm) in the phase diagram.
Cu-Zn
Eutectoid
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Eutectic and Eutectoid Reactions
Temperature
l
γ
Eutectoid
temperature
α
γ +l
l +β
Eutectic
temperature
γ +β
α+γ
β
α +β
Eutectoid
composition
Eutectic
composition
Composition
The above phase diagram contains both an eutectic reaction
and its solid-state analog, an eutectoid reaction
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Peritectic Reactions
A peritectic reaction - solid phase and liquid phase will
together form a second solid phase at a particular
temperature and composition upon cooling, e.g. L + α ↔ β
Temperature
These reactions are rather slow as the product phase will
form at the boundary between the two reacting phases thus
separating them, and slowing down any further reaction.
α + liquid
α +β
β
liquid
β + liquid
Peritectics are not as common as eutectics and eutectiods,
but do occur in some alloy systems. There is one in the FeC system that we will consider later.
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Congruent phase transformations
A congruent transformation involves no change in
composition (e.g., allotropic transformation such as α-Fe to
γ-Fe or melting transitions in pure solids).
For an incongruent transformation, at least one phase
changes composition (e.g. eutectic, eutectoid, peritectic
reactions).
Congruent
melting of γ
MSE 2090: Introduction to Materials Science
Ni-Ti
Chapter 9, Phase Diagrams
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The Iron–Iron Carbide (Fe–Fe3C) Phase Diagram
In their simplest form, steels are alloys of Iron (Fe) and
Carbon (C). The Fe-C phase diagram is a fairly complex
one, but we will only consider the steel part of the diagram,
up to around 7% Carbon.
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Phases in Fe–Fe3C Phase Diagram
¾ α-ferrite - solid solution of C in BCC Fe
• Stable form of iron at room temperature.
• The maximum solubility of C is 0.022 wt%
• Transforms to FCC γ-austenite at 912 °C
¾ γ-austenite - solid solution of C in FCC Fe
• The maximum solubility of C is 2.14 wt %.
• Transforms to BCC δ-ferrite at 1395 °C
• Is not stable below the eutectoid temperature
(727 ° C) unless cooled rapidly (Chapter 10)
¾ δ-ferrite solid solution of C in BCC Fe
• The same structure as α-ferrite
• Stable only at high T, above 1394 °C
• Melts at 1538 °C
¾ Fe3C (iron carbide or cementite)
• This intermetallic compound is metastable, it
remains as a compound indefinitely at room T, but
decomposes (very slowly, within several years)
into α-Fe and C (graphite) at 650 - 700 °C
¾ Fe-C liquid solution
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A few comments on Fe–Fe3C system
C is an interstitial impurity in Fe. It forms a solid solution
with α, γ, δ phases of iron
Maximum solubility in BCC α-ferrite is limited (max.
0.022 wt% at 727 °C) - BCC has relatively small interstitial
positions
Maximum solubility in FCC austenite is 2.14 wt% at 1147
°C - FCC has larger interstitial positions
Mechanical properties: Cementite is very hard and brittle can strengthen steels. Mechanical properties also depend
on the microstructure, that is, how ferrite and cementite are
mixed.
Magnetic properties: α -ferrite is magnetic below 768 °C,
austenite is non-magnetic
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Classification. Three types of ferrous alloys
¾ Iron: less than 0.008 wt % C in α−ferrite at room T
¾ Steels: 0.008 - 2.14 wt % C (usually < 1 wt % )
α-ferrite + Fe3C at room T
Examples of tool steel (tools for cutting other metals):
Fe + 1wt % C + 2 wt% Cr
Fe + 1 wt% C + 5 wt% W + 6 wt % Mo
Stainless steel (food processing equipment, knives,
petrochemical equipment, etc.): 12-20 wt% Cr, ~$1500/ton
¾ Cast iron: 2.14 - 6.7 wt % (usually < 4.5 wt %)
heavy equipment casing
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Eutectic and eutectoid reactions in Fe–Fe3C
Eutectic: 4.30 wt% C, 1147 °C
L ↔ γ + Fe3C
Eutectoid: 0.76 wt%C, 727 °C
γ(0.76 wt% C) ↔ α (0.022 wt% C) + Fe3C
Eutectic and eutectoid reactions are very important in
heat treatment of steels
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Development of Microstructure in Iron - Carbon alloys
Microstructure depends on composition (carbon
content) and heat treatment. In the discussion below we
consider slow cooling in which equilibrium is maintained.
Microstructure of eutectoid steel (I)
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Microstructure of eutectoid steel (II)
When alloy of eutectoid composition (0.76 wt % C) is
cooled slowly it forms perlite, a lamellar or layered
structure of two phases: α-ferrite and cementite (Fe3C)
The layers of alternating phases in pearlite are formed for
the same reason as layered structure of eutectic structures:
redistribution C atoms between ferrite (0.022 wt%) and
cementite (6.7 wt%) by atomic diffusion.
Mechanically, pearlite has properties intermediate to soft,
ductile ferrite and hard, brittle cementite.
In the micrograph, dark areas are
Fe3C layers, light phase is α-ferrite
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Microstructure of hypoeutectoid steel (I)
Compositions to the left of eutectoid (0.022 - 0.76 wt % C)
hypoeutectoid (less than eutectoid -Greek) alloys.
γ → α + γ → α + Fe3C
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Microstructure of hypoeutectoid steel (II)
Hypoeutectoid alloys contain proeutectoid ferrite (formed
above the eutectoid temperature) plus the eutectoid perlite
that contain eutectoid ferrite and cementite.
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Microstructure of hypereutectoid steel (I)
Compositions to the right of eutectoid (0.76 - 2.14 wt % C)
hypereutectoid (more than eutectoid -Greek) alloys.
γ → γ + Fe3C → α + Fe3C
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Microstructure of hypereutectoid steel
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Microstructure of hypereutectoid steel (II)
Hypereutectoid alloys contain proeutectoid cementite
(formed above the eutectoid temperature) plus perlite that
contain eutectoid ferrite and cementite.
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How to calculate the relative amounts of proeutectoid
phase (α or Fe3C) and pearlite?
Application of the lever rule with tie line that extends from
the eutectoid composition (0.75 wt% C) to α – (α + Fe3C)
boundary (0.022 wt% C) for hypoeutectoid alloys and to (α
+ Fe3C) – Fe3C boundary (6.7 wt% C) for hipereutectoid
alloys.
Fraction of α phase is determined by application of the
lever rule across the entire (α + Fe3C) phase field.
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Example for hypereutectoid alloy with composition C1
Fraction of pearlite:
WP = X / (V+X) = (6.7 – C1) / (6.7 – 0.76)
Fraction of proeutectoid cementite:
WFe3C = V / (V+X) = (C1 – 0.76) / (6.7 – 0.76)
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The Gibbs phase rule
Let’s consider a simple one-component system.
In the areas where only one
phase is stable both pressure
and temperature can be
independently varied without
upsetting
the
phase
equilibrium → there are 2
degrees of freedom.
P
liquid
solid
gas
T
Along the lines where two phases coexist in equilibrium,
only one variable can be independently varied without
upsetting the two-phase equilibrium (P and T are related by
the Clapeyron equation) → there is only one degree of
freedom.
At the triple point, where solid liquid and vapor coexist any
change in P or T would upset the three-phase equilibrium
→ there are no degrees of freedom.
In general, the number of degrees of freedom, F, in a
system that contains C components and can have Ph phases
is given by the Gibbs phase rule:
F = C − Ph + 2
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The Gibbs phase rule – example of a binary system
F = C − Ph + 2
P = const
F = C − Ph + 1
C=2
F = 3 − Ph
¾ In one-phase regions of the phase diagram T and XB can
be changed independently.
¾ In two-phase regions, F = 1. If the temperature is
chosen independently, the compositions of both
phases are fixed.
¾ Three phases (L, α, β) are in equilibrium only at the
eutectic point in this two component system (F = 0).
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Summary
Make sure you understand language and concepts:
¾ Austenite
¾ Cementite
¾ Component
¾ Congruent transformation
¾ Equilibrium
¾ Eutectic phase
¾ Eutectic reaction
¾ Eutectic structure
¾ Eutectoid reaction
¾ Ferrite
¾ Gibbs phase rule
¾ Hypereutectoid alloy
¾ Hypoeutectoid alloy
¾ Intermediate solid solution
¾ Intermetallic compound
¾ Invariant point
¾ Isomorphous
¾ Lever rule
¾ Liquidus line
¾ Metastable
¾
¾
¾
¾
¾
¾
¾
¾
¾
¾
¾
¾
¾
¾
¾
MSE 2090: Introduction to Materials Science
Chapter 9, Phase Diagrams
Microconstituent
Pearlite
Peritectic reaction
Phase
Phase diagram
Phase equilibrium
Primary phase
Proeutectoid cementite
Proeutectoid ferrite
Solidus line
Solubility limit
Solvus line
System
Terminal solid solution
Tie line
57